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Theorem ndmovg 7590
Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
Assertion
Ref Expression
ndmovg ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem ndmovg
StepHypRef Expression
1 df-ov 7412 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 eleq2 2823 . . . . . 6 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)))
3 opelxp 5713 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴𝑅𝐵𝑆))
42, 3bitrdi 287 . . . . 5 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑅𝐵𝑆)))
54notbid 318 . . . 4 (dom 𝐹 = (𝑅 × 𝑆) → (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ¬ (𝐴𝑅𝐵𝑆)))
6 ndmfv 6927 . . . 4 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
75, 6syl6bir 254 . . 3 (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴𝑅𝐵𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅))
87imp 408 . 2 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
91, 8eqtrid 2785 1 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  c0 4323  cop 4635   × cxp 5675  dom cdm 5677  cfv 6544  (class class class)co 7409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  ndmov  7591  curry1val  8091  curry2val  8095  1div0  11873  repsundef  14721  cshnz  14742  mamufacex  21891  mavmulsolcl  22053  mavmul0g  22055  iscau2  24794  1div0apr  29721  rrxsphere  47434
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